Wednesday, November 30, 2016

A Simple, Purely Geometric Proof that the Square Root of Two is Irrational—With a Couple of Bonus Side-Trips

Lots of proofs are based on simple ideas, but get bogged down in notation or exposition that doesn't bring out the salient points, as if the soloists and all the members of the choir were singing at equal volume.

Sometimes some warm-ups can help people understand what is essential in a proof and what is extra. They can help the simplicity of an idea shine through.

So the first warm-up is a purely geometric proof that the golden ratio is irrational. A number of years ago, I saw such a proof... but the diagram that went with it was laid out on a single line, and it got bogged down in a bunch of notation, and I kind of got it but it certainly didn't excite me.

Then one day I was looking at my business card and I realized that the proof was right there.  When I was designing my business card, I tried to figure out a good logo, and I eventually settled on a golden rectangle and golden spiral: